Recall that if \(\nabla A\in L^\infty (\mathbb {R}^d)\) , then the \(L^p(\mathbb {R}^d)\)  boundedness of \(T_{\Omega ,\,A}\)  follows by using the methods of rotation in the nice work of Caldéron [1]}, Bainshansky and Coifman [2]}. Since the method of rotations doesn't work in the case of \(\nabla A\in {\rm BMO}(\mathbb {R}^d)\) , Cohen [3]} and Hu [4]} obtained the \(L^p(\mathbb {R}^d)\)  boundedness of \(T_{\Omega ,\,A}\)  with smooth kernels by means of a good-\(\lambda \)  inequality. More precisly, if \(\Omega \in {\rm Lip}_\alpha (\mathbb {S}^{d-1})\)  (\(0<\alpha \le 1\) ), then Cohen [3]} proved that \(T_{\Omega ,A}\)  is a bounded operator on \(L^p(\mathbb {R}^d)\)  for \(1<p<\infty \) . Later on,
the result of Cohen [3]} was improved by Hofmann [7]}. It was shown that \(\Omega \in \cup _{q>1}L^q(\mathbb {S}^{d-1})\)  is a sufficient condition for the \(L^p(\mathbb {R}^d)\)  boundedness of \(T_{\Omega , A}\) . If \(\Omega \in L^{\infty }(\mathbb {S}^{d-1})\)  and \(w\in A_p(\mathbb {R}^d)\) , Hofmann further demonstrated that \(T_{\Omega ,A}\)  is bounded on \(L^p(\mathbb {R}^d,\,w)\) .
